JournalscmhVol. 86, No. 2pp. 317–351

Modular elliptic directions with complex multiplication (with an application to Gross’s elliptic curves)

  • Josep González Rovira

    Universitat Politècnica de Catalunya, Vilanova I La Geltrú, Spain
  • Joan-Carles Lario

    Universitat Politècnica de Catalunya, Barcelona, Spain
Modular elliptic directions with complex multiplication (with an application to Gross’s elliptic curves) cover
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Abstract

Let AfA_f be the abelian variety attached by Shimura to a normalized newform fS2(Γ1(N))f\in S_2(\Gamma_1(N)) and assume that AfA_f has elliptic quotients. The paper deals with the determination of the one dimensional subspaces (elliptic directions) in S2(Γ1(N))S_2(\Gamma_1(N)) corresponding to the pullbacks of the regular differentials of all elliptic quotients of AfA_f. For modular elliptic curves over number fields without complex multiplication (CM), the directions were studied by the authors in [8]. The main goal of the present paper is to characterize the directions corresponding to elliptic curves with CM. Then we apply the results obtained to the case N=p2N=p^2, for primes p>3p>3 and p3p\equiv 3 mod 44. For this case we prove that if ff has CM, then all optimal elliptic quotients of AfA_f are also optimal in the sense that its endomorphism ring is the maximal order of Q(p)\mathbb{Q}(\sqrt{-p}). Moreover, if ff has trivial Nebentypus then all optimal quotients are Gross’s elliptic curve A(p)A(p) and its Galois conjugates. Among all modular parametrizations J0(p2)A(p)J_0(p^2)\to A(p), we describe a canonical one and discuss some of its properties.

Cite this article

Josep González Rovira, Joan-Carles Lario, Modular elliptic directions with complex multiplication (with an application to Gross’s elliptic curves). Comment. Math. Helv. 86 (2011), no. 2, pp. 317–351

DOI 10.4171/CMH/225